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Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities. (English) Zbl 1125.34024

The authors introduce general oscillation criteria for the second order ordinary differential equation

p (t) x ' ' +q(t)x+ i=1 n q i (t)|x| α 1 sgn x=e(t),

where p,q,q i ,eC[0,), p(t)>0 and differentiable (but this assumption on the existence of p ' (t) is apparently not needed), α 1 >>α m >1>α m+1 >>α n , and no restriction is invoked on the forcing term e(t). Note that the equation contains both sublinear and superlinear terms due to the assumptions on the exponents α i . The main results (Theorems 1–3) are derived via the Riccati technique and generalize several results in the literature.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory